An improved bound for the Manickam-Mikl\'os-Singhi conjecture

TL;DR

This paper improves the lower bound on the number of non-negative sum subsets in a set of real numbers, advancing the understanding of the Manickam-Miklós-Singhi conjecture.

Contribution

It establishes a significantly stronger bound on n for the conjecture, surpassing previous results from 1987.

Findings

New bound: n > k(4e log k)^k ensures the conjecture holds.

At least inom{n-1}{k-1} subsets have non-negative sum under the new bound.

The result substantially advances the known conditions for the conjecture.

Abstract

We show that for $n>k(4e\log k)^k$ every set $\{x_1,..., x_n\}$ of $n$ real numbers with $\sum_{i=0}^{n}x_i \geq 0$ has at least $\binom{n-1}{k-1}$ $k$-element subsets of a non-negative sum. This is a substantial improvement on the best previously known bound of $n>(k-1)(k^k+k^2)+k$, proved by Manickam and Mikl\'os \cite{MM} in 1987.